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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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Objectives

– Recognize characteristics of parabolas
– Graph parabolas
– Determine a quadratic function’s minimum or
maximum value.

f(x)=ax2 +bx + c

graph to be a parabola.
Parabola

• Parabolas are
symmetric.

• Axis of Symmetry:
The line through
which the
parabola is
symmetric. Special Factored Form of the

• The vertex of the parabola is at (h,k)
and “a” describes the “steepness”
and direction of the parabola given by
the form

f (x) = a(x − h)2 + k
Minimum or maximum values of a function
occur at the VERTEX.

 P(x) = a(x – h)2 + k Vertex of parabola = (h, k)

a > 0 parabola opens up (h,k) = minimum point
Minimum Value of function is P(h)=k

a < 0 parabola opens down (h,k)=maximum point
Maximum Value of function is P(h)=k

Minimum (or maximum) function value for
a quadratic occurs at the vertex.

• If parabola opens up, f(x) has a
minimum value.

• If it opens down, f(x) has a maximum
value.

• Minimum/Maximum values are based on y-values.
Graph of f (x) = 2x2 − 4x + 3 = 2(x −1)2 Vertex Formula

P(x) = ax2 + bx + c (a ≠ 0)

The following formula will give you the x-value
for the vertex of a quadratic: Coordinates of vertex: Example

• Determine the following for f(x) without
graphing.
• f(x) = -3(x – 2)² + 12

a.) Find the vertex.
b.) Find the equation of the axis of symmetry.
c.) Does f(x) open up or down?
d.) Does f(x) have a max. or min. value?
Where does this value occur?
e.) What is the domain of f(x)?
f.) What is the range of f(x)?
Example

• Determine the following for f(x) without
graphing.
• f(x) = 2x² - 8 x - 3

a.) Find the vertex.
b.) Find the equation of the axis of symmetry.
c.) Does f(x) open up or down?
d.) Does f(x) have a max. or min. value?
Where does this value occur?
e.) What is the domain of f(x)?
f.) What is the range of f(x)?

1. Find the coordinates of the vertex.
(Use the vertex formula.)

2. Determine which way parabola opens by looking at a.
a > 0 parabola opens up (Vertex is lowest point)
a < 0 parabola opens down (Vertex is highest point)

3. Find the x-intercept(s). (Set y = 0)

4. Find the y-intercept. (Set x = 0)

2. Graph additional points if needed by t-chart or symmetry.
Use the vertex and
intercepts to graph
f(x) = 5 – 4x – x²

Give the equation
of the axis of
symmetry.

Determine the
domain and the
range of f(x). 